Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf $\mathcal{F}$ endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition $\mathcal{F}\simeq \mathcal{U}\oplus \mathcal{A}$ where $\mathcal{U}$ is a hermitian flat bundle and $\mathcal{A}$ is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles $f_{\ast }{\omega }_{X\mathrm{/}Y}^{\otimes m}$ under a surjective morphism $f\text{\hspace{0.17em}⁣}:X\to Y$ of smooth projective varieties with $m\ge 2$. This extends previous results of Fujita, Catanese–Kawamata, and Iwai.